Positivity 11 (2007), 69–75
2006 Birkh¨auser Verlag Basel/Switzerland
1385-1292/010069-7, published online October 13, 2006
A Reduction Theorem for Capacity
of Positive Maps
Abstract. We prove a reduction theorem for capacity of positive unital maps
of ﬁnite dimensional C
−algebras, thus reducing the computation of capacity
to the case when the image of a nonscalar projection is never a projection.
Mathematics Subject Classiﬁcation (2000). 81P68, 46N50.
Keywords. Quantum computing, capacity, positive maps.
In quantum information theory there has been a great deal of interest in the
concept of capacity of completely positive maps. A drawback with capacity is that
it is usually quite diﬃcult to compute, hence there is a need for developing com-
putational techniques. In the present paper we shall prove a reduction theorem for
capacity which reduces its computation to the ergodic case. As a consequence we
get a partial result towards the additivity of capacity for tensor products.
If P is a ﬁnite dimensional C
−algebra we denote by Tr
the trace on P
which takes the value 1 at each minimal projection. Let η denote the real func-
tion η(t)=−t log t for t>0, and η(0) = 0. Then the entropy S(a) of a positive
operator a in P is deﬁned by S(a)=Tr
(η(a)). If M is another ﬁnite dimen-
−algebra let Φ: M → P be a positive unital linear trace preserving map,
(Φ(x)) = Tr
(x) for all x ∈ M. Note that we only assume Φ is positive
and not completely positive, since the latter stronger assumption is in most cases
unnecessary. Let D denote the positive operators in M with trace 1. If a ∈ D let
where the sup is over all convex combinations of operators a
∈ D with
The capacity C(Φ) of Φ, or C
is deﬁned by
C(Φ) = sup
see e.g. section 3.2 in .